Group Theory

Theorem: The intersection of subgroups $H_1,H_2,....$ is a subgroup of each of $H_1,H_2,...$

We say the elements $g_1,...,g_m$ are independent if none of them can be expressed in terms of the others, that is, $g_i\notin ⟨g_1,...,g_{i-1},g_{i+1},...,g_m⟩$. Clearly every finite group has at least one set of independent generators. Independent elements can have relations between them, e.g. if $a,b$ are independent then we may have $(ab{)}^2=1$ for example. Such a relation is called a defining relation.

Given any two groups $G,H$ we may form their direct product $G\times H$, whose elements are pairs $(g,h)$ with $g\in G,h\in H$, and the group operation applies coordinatewise. The direct product of abelian groups is abelian.

Suppose every element of a group $F$ has the form $gh$ where $g\in G,h\in H$ for some subgroups $G,H$ of $F$, and furthermore, suppose every element of $G$ commutes with every element of $H$ and $G\cap H=\{1\}$. Then $F\cong G\times H$.

It is clear how to generalize this to define the direct product to $k$ groups.

Example: ${\mathbb{Z}}_{15}^{*}\cong {\mathbb{Z}}_4\times {\mathbb{Z}}_2$.